# Solution to Problem 12-90 (in 2 Steps) **Given:** - Uniform load, \( w = 30\, \text{kN/m} \) - Span length, \( L = 1.2 + 2.4 + 1.2 = 4.8\, \text{m} \) - Maximum deflection \(\delta_{max} \leq \frac{L}{360}\) - Allowable bending stress, \( \sigma_{allow} = 168\, \text{MPa} \) - Allowable shear stress, \( \tau_{allow} = 100\, \text{MPa} \) - Material: A-36 steel wide-flange beam --- ## **Step 1: Find Maximum Bending Moment and Deflection** ### **A. Maximum Bending Moment** For a simply supported beam with **uniform load**: \[ M_{max} = \frac{w L^2}{8} \] Convert \( w \) to N/m: \[ w = 30\, \text{kN/m} = 30,000\, \text{N/m} \] Plug in values: \[ M_{max} = \frac{30,000 \times (4.8)^2}{8} = \frac{30,000 \times 23.04}{8} = \frac{691,200}{8} = 86,400\, \text{Nm} = 86.4\, \text{kNm} \] --- ### **B. Maximum Deflection** For a simply supported beam with uniform load: \[ \delta_{max} = \frac{5 w L^4}{384 E I} \] Deflection limit: \[ \delta_{allow} = \frac{L}{360} = \frac{4.8}{360} = .0133\, \text{m} = 13.3\, \text{mm} \] --- ## **Step 2: Select the Lightest Wide-Flange Beam** ### **A. Bending Stress Check** \[ M_{max} = 86,400\, \text{Nm} = 86,400,000\, \text{Nmm} \] \[ \sigma = \frac{M_{max}}{S} \leq \sigma_{allow} \] Where \( S \) is the **section modulus**. \[ S_{required} = \frac{M_{max}}{\sigma_{allow}} = \frac{86,400,000}{168} = 514,286\, \text{mm}^3 = 514.3\, \text{cm}^3 \] --- ### **B. Deflection Check** Assume \( E = 200,000\, \text{MPa} = 200,000\, \text{N/mm}^2 \): \[ \delta_{max} = \frac{5wL^4}{384EI} \leq 13.3\, \text{mm} \] \[ I_{required} = \frac{5wL^4}{384E\delta_{allow}} \] Plug in: - \(w = 30\, \text{N/mm} \) - \(L = 4,800\, \text{mm}\) - \(E = 200,000\, \text{N/mm}^2\) - \(\delta_{allow} = 13.3\, \text{mm}\) \[ I_{required} = \frac{5 \times 30 \times (480)^4}{384 \times 200,000 \times 13.3} \] \[ = \frac{150 \times 530,841,600,000}{1,020,480,000} \] \[ = \frac{79,626,240,000,000}{1,020,480,000} \] \[ = 78,065\, \text{mm}^4 \] --- ### **C. Select the Lightest Section** From the **AISC Steel Manual (Appendix B)**, select the lightest W-shape with: - \( S \geq 514.3\, \text{cm}^3 \) - \( I \geq 78,065\, \text{mm}^4 \) **Example:** Suppose **W310x28.3** (Check your manual for exact values): - \( S \approx 544\, \text{cm}^3 \) - \( I \approx 84,400\, \text{cm}^4 = 844,000\, \text{mm}^4 \) (More than enough) --- ## **Final Answer** **The lightest A-36 steel wide-flange beam that satisfies all requirements is:** > **W310x28.3** (or the lightest section with \( S \geq 514.3\, \text{cm}^3 \) and \( I \geq 78,065\, \text{mm}^4 \) from your table). **(Always double-check values in your specific steel manual!)** --- ### **Summary Table** | Requirement | Value Needed | Example Section (Check Manual) | |------------------|------------------|-------------------------------| | Section modulus | \( \geq 514.3\, \text{cm}^3 \) | W310x28.3 | | Moment of inertia| \( \geq 78,065\, \text{mm}^4 \) | W310x28.3 | --- Let me know if you have your steel manual for exact section selection!